Why would my wife ask about "including" another woman if we can never agree on which one we would choose?

Just tell her it is a matter of personal opinion and that you won't criticize her choices if she will accord you the same courtesy. Then offer to have a calm nonjudgmental conversation about the various choices without it becoming hostile. And, if that doesn't work then just leave it alone completely, refuse to play the game with her anymore and if she askes why you stopped tell her that you would rather not play at all instead of fighting about it every time.